scholarly journals On the difference method for approximation of second order derivatives of a solution of Laplace’s equation in a rectangular parallelepiped

Filomat ◽  
2019 ◽  
Vol 33 (2) ◽  
pp. 633-643
Author(s):  
Adiguzel Dosiyev ◽  
Hediye Sarikaya

We present and justify finite difference schemes with the 14-point averaging operator for the second derivatives of the solution of the Dirichlet problem for Laplace?s equations on a rectangular parallelepiped. The boundary functions ?j on the faces ?j,j = 1,2,..., 6 of the parallelepiped are supposed to have fifth derivatives belonging to the H?lder classes C5?, 0 < ? < 1. On the edges, the boundary functions as a whole are continuous, and their second and fourth order derivatives satisfy the compatibility conditions which result from the Laplace equation. It is proved that the proposed difference schemes for the approximation of the pure and mixed second derivatives converge uniformly with order O(h3+?), 0 < ? < 1 and O(h3), respectively. Numerical experiments are illustrated to support the theoretical results.

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 893-901 ◽  
Author(s):  
Adiguzel Dosiyev ◽  
Ahlam Abdussalam

The boundary functions ?j of the Dirichlet problem, on the faces ?j, j = 1,2,..., 6 of the parallelepiped R are supposed to have seventh derivatives satisfying the H?lder condition and on the edges their second, fourth and sixth order derivatives satisfy the compatibility conditions which result from the Laplace equation. For the error uh-u of the approximate solution uh at each grid point (x1,x2,x3), a pointwise estmation O(?h6) is obtained, where ?= ?(x1,x2,x3) is the distance from the current grid point to the boundary of R; h is the grid step. The solution of difference problems constructed for the approximate values of the first and pure second derivatives converge with orders O(h6 ?ln h?) and O(h5+?), 0 < ? < 1, respectivly.


2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Jinsong Hu ◽  
Youcai Xu ◽  
Bing Hu ◽  
Xiaoping Xie

Two conservative finite difference schemes for the numerical solution of the initialboundary value problem of Rosenau-Kawahara equation are proposed. The difference schemes simulate two conservative quantities of the problem well. The existence and uniqueness of the difference solution are proved. It is shown that the finite difference schemes are of second-order convergence and unconditionally stable. Numerical experiments verify the theoretical results.


Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 791-800 ◽  
Author(s):  
Adiguzel Dosiyev ◽  
Hediye Sarikaya

A 14-point difference operator is used to construct finite difference problems for the approximation of the solution, and the first order derivatives of the Dirichlet problem for Laplace?s equations in a rectangular parallelepiped. The boundary functions ?j on the faces ?j, j = 1,2,...,6 of the parallelepiped are supposed to have pth order derivatives satisfying the H?lder condition, i.e., ?j ? Cp,?(?j), 0 < ? < 1, where p = {4,5}. On the edges, the boundary functions as a whole are continuous, and their second and fourth order derivatives satisfy the compatibility conditions which result from the Laplace equation. For the error uh - u of the approximate solution uh at each grid point (x1,x2,x3), ?uh-u?? c?p-4(x1,x2,x3)h4 is obtained, where u is the exact solution, ? = ? (x1, x2,x3) is the distance from the current grid point to the boundary of the parallelepiped, h is the grid step, and c is a constant independent of ? and h. It is proved that when ?j ? Cp,?, 0 < ? < 1, the proposed difference scheme for the approximation of the first derivative converges uniformly with order O(hp-1), p ? {4,5}.


1992 ◽  
Vol 70 (2) ◽  
pp. 443-449 ◽  
Author(s):  
Jerzy Cioslowski ◽  
Stacey T. Mixon

Nonbonding repulsive interactions between hydrogen atoms, separated by less than ca. 2.18 Å and connected to two carbon atoms in the 1,4-positions, are associated with additional pairs of bond and ring critical points in the electron density, ρ(r), and the corresponding attractor interaction lines. Such topological features of ρ(r) are present in some planar benzenoid hydrocarbons, including chrysene, benzanthracene, and phenanthrene. They also appear in conformations of the biphenyl molecule with the torsional angles between the benzene rings lying within the range of 0°–27°. Properties such as the bond point – ring point distance, the difference in ρ(r) at the bond point and the ring point, and the bond ellipticity are found to follow universal functional dependencies with the hydrogen–hydrogen distance as the controlling variable. The same is true about the corresponding difference in the second derivatives of ρ(r) in the direction of the vector connecting the bond and ring points. Keywords: atoms in molecules, electron density, attractor interaction lines, steric repulsions.


Author(s):  
Frederick P. Gardiner

SynopsisDual extremum problems associated with an infinite family of admissible loops on a Riemann surface are shown to be solvable by Jenkins-Strebel differentials. Then the inequalities associated with these problems are used to calculate the first derivatives of extremal length functionals on Teichmüller space and to estimate the difference quotients for the second derivatives of these functions.


2006 ◽  
Vol 16 (05) ◽  
pp. 679-699
Author(s):  
HOMARE MORIOKA ◽  
ATUSI TANI

This paper is devoted to study the global existence of a solution of bounded variation to the initial value problem for a system of conservation laws with artificial viscosity. The method of finite difference schemes of implicit type is used. We prove that the difference approximations converge to a weak solution of the problem. Moreover, this weak solution is really a classical solution according to the Oleĭnik's argument.


2018 ◽  
Vol 22 ◽  
pp. 01011
Author(s):  
Suzan Cival Buranay ◽  
Lawrence Adedayo Farinola

We construct four-point implicit difference boundary value problem for the first derivative of the solution u(x,t) of the first type boundary value problem for one dimensional heat equation with respect to the time variable t. Also, for the second derivatives of u(x,t) special four-point implicit difference boundary value problems are proposed. It is assumed that the initial function belongs to the Hölder space C8+α,0 < α < 1, the heat source function given in the heat equation is from the Hölder space [see formula in PDF], the boundary functions are from [see formula in PDF], and between the initial and the boundary functions the conjugation conditions of orders q = 0,1,2,3,4 are satisfied. We prove that the solution of the proposed difference schemes converge uniformly on the grids of the order O(h2+τ) (second order accurate in the spatial variable x and first order accurate in time t) where, h is the step size in x and τ is the step size in time. Theoretical results are justified by numerical examples.


2002 ◽  
Vol 2 (2) ◽  
pp. 132-142 ◽  
Author(s):  
Francisco J. Gaspar ◽  
Francisco J. Lisbona ◽  
Petr N. Vabishchevich

AbstractIn this paper, we present a finite difference analysis of the consolidation problem for saturated porous media. In the classical model, the behaviour of the porous environment – fluid system is described by a set of equations for the unknown vector displacements of the matrix skeleton and the fluid pressure. For simplicity we consider a model problem with constant coefficients in a rectangular domain. A priori estimates for the difference solution of the problem are obtained and on their basis the convergence of two-level difference schemes is investigated.


Author(s):  
N. K. Ashirbayev ◽  
◽  
Zh.N. Ashirbayeva ◽  
M.T. Shomanbayeva ◽  
R.B. Bekmoldayeva ◽  
...  

The work is devoted to the generalization of the difference method of spatial characteristics to the case of the plane problem of the propagation of waves in a rectangular region of finite dimensions with a symmetrically-located rectangular cutout at the lateral boundaries. Based on the numerical technique developed in this work, the calculated finite - difference relations of dynamic problems are obtained at the corner points of a rectangular cutout, where the smoothness of functions that is “familiar” to dynamic problems is violated. At these corner points, the first and second derivatives of the desired functions suffer a discontinuity of the first kind. The results of the study are brought to a numerical solution. The effect of stress concentration in the vicinity of the cutout was studied and it was shown that the impact of the cutout on the particle velocity distribution, on the stress distribution has a local character.


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